The One-Dimensional Schrödinger Equation with a Quasiperiodic Potential

Dinaburg, E. I.; Sinaĭ, Ya. G.

doi:10.1007/978-1-4419-6205-8_11

Cited by 45 publications

(252 citation statements)

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“…This approach was taken by Dinaburg & Sinai [17] and further developed by Eliasson [18] to show that for a large enough, the skew-product is reducible when the rotation number is either resonant or Diophantine with respect to ω. The averaging approach for this fast periodic forcing [39] works very well and one can obtain the asymptotics of the rotation number and the Lyapunov exponent at +∞ [16] together with an estimate on the decay of gap length [15].…”

Section: 2mentioning

confidence: 99%

Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies

Puig

Simó

2010

Regul. Chaot. Dyn.

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Abstract. In this paper we investigate numerically the following Hill's equation x ′′ + (a + bq(t))x = 0 where q(t) = cos t + cos √ 2t + cos √ 3t is a quasiperiodic forcing with three rationally independent frequencies. It appears,also, as the eigenvalue equation of a Schrödinger operator with quasi-periodic potential.Massive numerical computations were performed for the rotation number and the Lyapunov exponent in order to detect open and collapsed gaps, resonance tongues. Our results show that the quasi-periodic case with three independent frequencies is very different not only from the periodic analogs, but also from the case of two frequencies. Indeed, for large values of b the spectrum contains open intervals at the bottom. From a dynamical point of view we numerically give evidence of the existence of open intervals of a, for large b where the system is nonuniformly hyperbolic: the system does not have an exponential dichotomy but the Lyapunov exponent is positive. In contrast with the region with zero Lyapunov exponents, both the rotation number and the Lyapunov exponent do not seem to have square root behavior at endpoints of gaps. The rate of convergence to the rotation number and the Lyapunov exponent in the nonuniformly hyperbolic case is also seen to be different from the reducible case."But our tongues get out of control. They are restless and evil, and always spreading deadly poison". (James 3, 6) To our friend Henk Broer, who likes quasi-periodicity so much, on his 60th birthday.

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Section: 2mentioning

confidence: 99%

Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies

Puig

Simó

2010

Regul. Chaot. Dyn.

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“…have been found in [7,22,30,8] along with the presence of gaps in the spectrum. Actually, to any quasi-periodic potential q and any eigenvalue E * can be associated a rotation number…”

Section: Quasi-periodic Potentials: a Short Overview Of Known Resultsmentioning

confidence: 89%

“…For completeness, and even if this case doesn't exactly fit in our general framework 7 , we display here the numerical outcome of the algorithm written in [40] to solve the problem (13); it is just a special case of (11). Namely what we want to check is the ability of this numerical processing to recover well-known results, like e.g.…”

Section: A Numerical Visualization Of a Continuous "Almost-mathieu" Mmentioning

confidence: 99%

“…3 in [6] where periodic approximations are studied by means of a spectral scheme like in [15]; the corresponding potential is displayed on Fig. 2 in the same 7 It is better suited for 1-D quasicrystals. In order to provide some explanation about the figures A.1 and A.2, we turn back to the notion of "resonance tongue", see Definition 2 and [3,4].…”

Section: A Numerical Visualization Of a Continuous "Almost-mathieu" Mmentioning

confidence: 99%

“…In §2.1, we shall review briefly the rigorous mathematical results on eigenfunctions for quasi-periodic 1-D problems which followed the seminal article by Dinaburg and Sinai, [7]. They all rely on difficult KAM arguments and are mostly concerned with bounded solutions of the quasi-periodic Hill's equation y + (E − q(x)) = 0, for small or high E, see [30,22,11,40,8].…”

Section: Introductionmentioning

confidence: 99%

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Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal

Gosse¹

2008

Nonlinear Analysis: Real World Applications

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This paper investigates a simple one-dimensional model of incommensurate "harmonic crystal" in terms of the spectrum of the corresponding Schrödinger equation. Two angles of attack are studied: the first exploits techniques borrowed from the theory of quasi-periodic functions while the second relies on periodicity properties in a higher-dimensional space. It is shown that both approaches lead to essentially the same results; that is, the lower spectrum is splitted between "Cantor-like zones" and "impurity bands" to which correspond critical and extended eigenstates, respectively. These "new bands" seem to emerge inside the band gaps of the unperturbed problem when certain conditions are met and display a parabolic nature. Numerical tests are extensively performed on both steady and time-dependent problems.Key words: Quasi-periodic potential, impurity band, quasi-Bloch wave, quantum chaos, spectral algorithm, deformed crystal, Charge-Density Wave (CDW). 1991 MSC: 81Q05, 81Q20 IntroductionThis article is a step further in the numerical study of electronic motion in a one-dimensional model of "harmonic crystal", that is to say, a modeling in solid-state physics going beyond the rigid periodic lattice picture. Indeed, it consists in assuming that, within the BornOppenheimer approximation (the motion of nucleons is much slower than the electron's ones and thus decouples adiabatically), ionic cores are linked together by springs and vibrate like classical coupled oscillators; consult the book [1] and the recent paper [14] to which we are about to borrow notation.In [14], the following rescaled problem describing the motion of a free electron inside a 1-D lattice of atoms which display single-mode vibrations has been investigated within the * Corresponding Author Email address: l.gosse@ba.iac.cnr.it (Laurent Gosse). Preprint submitted to Elsevier ScienceWKB semi-classical framework [5,19] for the following Schrödinger equation,and under the fundamental assumption that k and Ω(k) = | sin(kπ)| are rational and commensurate. This is perhaps the simplest non-trivial model available which describes the effects of phonons onto electronic conduction and is sometimes referred to as the "modulated crystal" [1,9,25,26,32]. Since the potential in (1) isn't of the form cos(x − 1 2 at 2 ), the unitary transform propsed in [35] and leading to Bloch oscillations doesn't apply here (see also [33] for an AC-type modulation). A particular feature pointed out in [39] and called lattice tracking which refers to the dragging effect of heavy nucleons onto the much lighter electron was easily observed in the numerical results.The commensurability hypothesis is of critical importance in this global picture in order to keep on applying the classic Bloch decomposition for (1); we assume the reader familiar with this theory and refer only to [1,5,15,19] for details. The first step in this line of thinking is the derivation of "energy bands", which are dispersion relations for electrons around certain levels of excitation. Solids are clas...

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Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schrödinger Equation

Zhao

2016

Commun. Math. Phys.

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The One-Dimensional Schrödinger Equation with a Quasiperiodic Potential

Cited by 45 publications

References 3 publications

Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies

Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies

Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal

Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schrödinger Equation

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